Subgroups — C₅×#83531 #14459103

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#547	C₂	1	1	yes	maximal, characteristic
2	C₅×D₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5094	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5041	C₂	1	2	yes	maximal, complemented
6	C₅×#5040	C₂	1	2	yes	maximal, complemented
7	C₅×#5019	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₅×#214	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4971	C₂	1	2	yes	maximal, complemented
11	C₂×C₅×#810	C₂	1	2	yes	maximal, complemented
12	C₅×#3298	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×#41	—	2	1	no	—
14	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#123	—	2	2	no	—
17	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
18	C₃×C₅×#121	C₂²	1	2	yes	—
19	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₅×D₄×Dic₃	—	2	1	no	—
23	C₂×C₅×#214	C₂²	1	1	yes	characteristic
24	C₂×C₅×#214	C₂²	1	2	yes	—
25	C₅×#924	C₂²	1	2	yes	—
26	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
27	C₅×#888	—	2	1	no	—
28	C₅×#878	—	2	2	no	—
29	C₅×#869	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
31	C₅×#832	C₂²	1	1	yes	characteristic
32	C₅×#830	C₂²	1	2	yes	—
33	C₅×#810	C₂²	1	1	yes	characteristic
34	C₅×#810	C₂²	1	2	yes	—
35	C₅×#810	C₂²	1	2	yes	—
36	C₅×#810	C₂²	1	2	yes	—
37	C₅×#810	C₂²	1	2	yes	—
38	C₅×#808	C₂²	1	1	yes	characteristic
39	C₅×#808	C₂²	1	2	yes	—
40	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
42	#83531	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×D₄×Q₈	—	3	1	no	—
44	C₅×#547	D₃	1	1	yes	characteristic
45	C₅×#487	—	3	1	no	—
46	C₅×#483	—	3	2	no	—
47	C₅×#475	—	3	1	no	—
48	C₅×#465	—	3	2	no	—
49	C₅×#461	—	3	1	no	—
50	C₅×#448	—	3	2	no	—
51	C₄×C₅×SD₁₆	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₂×C₅×#103	—	3	2	no	—
54	C₃×C₅×#41	—	4	4	no	—
55	C₂×C₃×C₅×Q₈	—	2	1	no	—
56	C₂×C₃×C₅×D₄	—	2	2	no	—
57	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
58	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	—	2	2	no	—
60	C₂²×C₃×C₄×C₅	—	2	1	no	—
61	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
62	C₂²×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₆	—	2	4	no	—
64	C₂×C₅×Dic₆	—	2	2	no	—
65	C₂×C₅×Dic₆	—	4	2	no	—
66	C₂×C₅×Dic₆	C₂³	1	2	yes	—
67	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
68	C₃×C₅×C₄⋊C₄	—	2	2	no	—
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
73	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
75	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
76	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
77	C₅×#217	—	2	2	no	—
78	C₅×#214	—	2	2	no	—
79	C₅×#214	—	4	2	no	—
80	C₅×C₄⋊Dic₃	—	2	1	no	—
81	C₅×C₄⋊Dic₃	—	2	1	no	—
82	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
83	C₅×Dic₃⋊C₄	—	2	1	no	—
84	C₅×Dic₃⋊C₄	—	2	2	no	—
85	C₄×C₅×Dic₃	—	2	1	no	—
86	C₂×C₅×C₃⋊C₈	—	4	2	no	—
87	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
89	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
90	C₃×#547	C₂×C₅	1	1	yes	characteristic
91	D₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
92	#5094	C₂×C₅	1	1	yes	characteristic, complemented
93	#5093	C₂×C₅	1	1	yes	characteristic, complemented
94	#5041	C₂×C₅	1	2	yes	complemented
95	#5040	C₂×C₅	1	2	yes	complemented
96	#5019	C₂×C₅	1	1	yes	characteristic, complemented
97	C₄×#214	C₂×C₅	1	1	yes	characteristic, complemented
98	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
99	#4971	C₂×C₅	1	2	yes	complemented
100	C₂×#810	C₂×C₅	1	2	yes	complemented
101	C₂×C₅×#41	—	2	1	no	—
102	C₂²×C₅×Q₈	—	3	2	no	—
103	C₂×C₅×SD₁₆	—	3	1	no	—
104	C₂×C₅×SD₁₆	—	3	2	no	—
105	C₂²×C₅×C₈	—	3	2	no	—
106	C₅×C₄⋊Q₈	—	3	1	no	—
107	C₅×C₄⋊Q₈	—	6	1	no	—
108	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
109	C₅×#123	—	2	2	no	—
110	C₅×C₂²⋊Q₈	—	2	1	no	—
111	C₅×C₂²⋊Q₈	—	3	2	no	—
112	C₅×C₂²⋊Q₈	—	6	2	no	—
113	C₅×#121	C₂×D₃	1	2	yes	—
114	C₄×C₅×Q₈	—	3	1	no	—
115	C₄×C₅×D₄	—	6	1	no	—
116	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
117	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
118	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
119	C₅×#106	—	3	1	no	—
120	C₅×#106	—	3	2	no	—
121	C₅×C₄⋊C₈	—	3	1	no	—
122	C₅×#103	—	3	1	no	—
123	C₅×#103	—	3	2	no	—
124	C₅×#103	—	3	2	no	—
125	C₅×#103	—	3	2	no	—
126	C₅×#103	—	3	2	no	—
127	C₅×#102	—	3	1	no	—
128	C₅×C₂²⋊C₈	—	3	2	no	—
129	C₄×C₅×C₈	—	3	1	no	—
130	#3298	—	3	1	no	Sylow
131	C₂³×C₃×C₅	—	2	1	no	—
132	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
133	C₃×C₅×Q₈	—	4	2	no	—
134	C₃×C₅×D₄	—	2	1	no	—
135	C₃×C₅×D₄	—	2	1	no	—
136	C₃×C₅×D₄	—	4	4	no	—
137	C₃×C₅×D₄	SD₁₆	1	4	yes	complemented
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	2	2	no	—
140	C₂×C₃×C₄×C₅	—	2	1	no	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	—	4	4	no	—
143	C₂×C₃×C₄×C₅	—	4	2	no	—
144	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
148	C₂×C₅×Dic₃	—	2	1	no	—
149	C₂×C₅×Dic₃	—	2	1	no	—
150	C₂×C₅×Dic₃	—	2	2	no	—
151	C₂×C₅×Dic₃	—	4	4	no	—
152	C₅×Dic₆	—	2	1	no	—
153	C₅×Dic₆	—	2	2	no	—
154	C₅×Dic₆	—	2	2	no	—
155	C₅×Dic₆	—	4	4	no	—
156	C₅×Dic₆	#41	1	2	yes	complemented
157	C₅×C₃⋊C₈	—	2	1	no	—
158	C₅×C₃⋊C₈	—	2	1	no	—
159	C₅×C₃⋊C₈	—	2	2	no	—
160	C₂×C₃×#41	—	2	1	no	—
161	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
162	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
163	C₃×#123	—	2	2	no	—
164	C₃×C₂²⋊Q₈	—	2	1	no	—
165	C₃×#121	C₂²×C₅	1	2	yes	—
166	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
167	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
168	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
169	D₄×Dic₃	—	2	1	no	—
170	C₂×#214	C₂²×C₅	1	1	yes	characteristic
171	C₂×#214	C₂²×C₅	1	2	yes	—
172	#924	C₂²×C₅	1	2	yes	—
173	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
174	#888	—	2	1	no	—
175	#878	—	2	2	no	—
176	#869	C₂²×C₅	1	1	yes	characteristic
177	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
178	#832	C₂²×C₅	1	1	yes	characteristic
179	#830	C₂²×C₅	1	2	yes	—
180	#810	C₂²×C₅	1	1	yes	characteristic
181	#810	C₂²×C₅	1	2	yes	—
182	#810	C₂²×C₅	1	2	yes	—
183	#810	C₂²×C₅	1	2	yes	—
184	#810	C₂²×C₅	1	2	yes	—
185	#808	C₂²×C₅	1	1	yes	characteristic
186	#808	C₂²×C₅	1	2	yes	—
187	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
188	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
189	C₅×#41	—	4	4	no	—
190	C₂×C₅×Q₈	—	2	1	no	—
191	C₂×C₅×Q₈	—	3	1	no	—
192	C₂×C₅×Q₈	—	3	2	no	—
193	C₂×C₅×Q₈	—	6	4	no	—
194	C₂×C₅×Q₈	—	6	2	no	—
195	C₂×C₅×Q₈	—	12	2	no	—
196	C₂×C₅×D₄	—	2	2	no	—
197	C₂×C₅×D₄	#67	1	1	yes	characteristic
198	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
199	C₂²×C₄×C₅	—	2	2	no	—
200	C₂²×C₄×C₅	—	2	1	no	—
201	C₂²×C₄×C₅	—	6	2	no	—
202	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
203	C₅×SD₁₆	—	6	2	no	—
204	C₅×SD₁₆	—	12	2	no	—
205	C₂×C₅×C₈	—	3	1	no	—
206	C₂×C₅×C₈	—	3	2	no	—
207	C₂×C₅×C₈	—	3	1	no	—
208	C₂×C₅×C₈	—	12	2	no	—
209	C₅×C₄⋊C₄	—	2	2	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	3	1	no	—
212	C₅×C₄⋊C₄	—	6	1	no	—
213	C₅×C₄⋊C₄	—	6	1	no	—
214	C₅×C₄⋊C₄	—	6	2	no	—
215	C₅×C₄⋊C₄	—	6	1	no	—
216	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
217	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
218	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
219	C₅×C₂²⋊C₄	—	2	1	no	—
220	C₅×C₂²⋊C₄	—	2	2	no	—
221	C₅×C₂²⋊C₄	—	6	2	no	—
222	C₅×C₂²⋊C₄	#67	1	2	yes	—
223	C₄²×C₅	—	6	1	no	—
224	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
225	D₄×Q₈	—	3	1	no	—
226	#547	C₅×D₃	1	1	yes	characteristic
227	#487	—	3	1	no	—
228	#483	—	3	2	no	—
229	#475	—	3	1	no	—
230	#465	—	3	2	no	—
231	#461	—	3	1	no	—
232	#448	—	3	2	no	—
233	C₄×SD₁₆	—	3	1	no	—
234	C₈×D₄	—	3	1	no	—
235	C₂×#103	—	3	2	no	—
236	C₂²×C₃×C₅	—	2	2	no	—
237	C₂²×C₃×C₅	—	2	2	no	—
238	C₂²×C₃×C₅	—	4	1	no	—
239	C₂²×C₃×C₅	—	4	1	no	—
240	C₂²×C₃×C₅	—	4	1	no	—
241	C₂²×C₃×C₅	C₂×SD₁₆	1	4	yes	—
242	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
243	C₃×C₄×C₅	—	2	2	no	—
244	C₃×C₄×C₅	—	2	1	no	—
245	C₃×C₄×C₅	—	4	1	no	—
246	C₃×C₄×C₅	—	4	2	no	—
247	C₃×C₄×C₅	C₂×SD₁₆	1	2	yes	—
248	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
249	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
250	C₅×Dic₃	—	2	2	no	—
251	C₅×Dic₃	—	4	2	no	—
252	C₅×Dic₃	—	4	1	no	—
253	C₃×#41	—	4	4	no	—
254	C₂×C₃×Q₈	—	2	1	no	—
255	C₂×C₃×D₄	—	2	2	no	—
256	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
257	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
258	C₂²×C₃×C₄	—	2	2	no	—
259	C₂²×C₃×C₄	—	2	1	no	—
260	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
261	C₂²×Dic₃	—	2	2	no	—
262	C₂×Dic₆	—	2	4	no	—
263	C₂×Dic₆	—	2	2	no	—
264	C₂×Dic₆	—	4	2	no	—
265	C₂×Dic₆	C₂³×C₅	1	2	yes	—
266	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
267	C₃×C₄⋊C₄	—	2	2	no	—
268	C₃×C₄⋊C₄	—	2	1	no	—
269	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
270	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
271	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
272	C₃×C₂²⋊C₄	—	2	1	no	—
273	C₃×C₂²⋊C₄	—	2	2	no	—
274	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
275	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
276	#217	—	2	2	no	—
277	#214	—	2	2	no	—
278	#214	—	4	2	no	—
279	C₄⋊Dic₃	—	2	1	no	—
280	C₄⋊Dic₃	—	2	1	no	—
281	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
282	Dic₃⋊C₄	—	2	1	no	—
283	Dic₃⋊C₄	—	2	2	no	—
284	C₄×Dic₃	—	2	1	no	—
285	C₂×C₃⋊C₈	—	4	2	no	—
286	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
287	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
288	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
289	C₂³×C₅	—	2	1	no	—
290	C₂³×C₅	C₂×#67	1	2	yes	—
291	C₅×Q₈	—	3	2	no	—
292	C₅×Q₈	—	4	2	no	—
293	C₅×Q₈	—	6	1	no	—
294	C₅×Q₈	—	6	2	no	—
295	C₅×Q₈	—	6	2	no	—
296	C₅×Q₈	—	12	4	no	—
297	C₅×D₄	—	2	1	no	—
298	C₅×D₄	—	2	1	no	—
299	C₅×D₄	—	4	4	no	—
300	C₅×D₄	#214	1	4	yes	complemented
301	C₂×C₄×C₅	—	2	2	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	4	4	no	—
306	C₂×C₄×C₅	—	4	2	no	—
307	C₂×C₄×C₅	—	6	1	no	—
308	C₂×C₄×C₅	—	6	1	no	—
309	C₂×C₄×C₅	—	6	2	no	—
310	C₂×C₄×C₅	—	12	4	no	—
311	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
312	C₂×C₄×C₅	C₂×#67	1	2	yes	—
313	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
315	C₅×C₈	—	6	1	no	—
316	C₅×C₈	—	6	1	no	—
317	C₅×C₈	—	6	2	no	—
318	C₂×#41	—	2	1	no	—
319	C₂²×Q₈	—	3	2	no	—
320	C₂×SD₁₆	—	3	1	no	—
321	C₂×SD₁₆	—	3	2	no	—
322	C₂²×C₈	—	3	2	no	—
323	C₄⋊Q₈	—	3	1	no	—
324	C₄⋊Q₈	—	6	1	no	—
325	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
326	#123	—	2	2	no	—
327	C₂²⋊Q₈	—	2	1	no	—
328	C₂²⋊Q₈	—	3	2	no	—
329	C₂²⋊Q₈	—	6	2	no	—
330	#121	C₂×C₅×D₃	1	2	yes	—
331	C₄×Q₈	—	3	1	no	—
332	C₄×D₄	—	6	1	no	—
333	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
334	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
336	#106	—	3	1	no	—
337	#106	—	3	2	no	—
338	C₄⋊C₈	—	3	1	no	—
339	#103	—	3	1	no	—
340	#103	—	3	2	no	—
341	#103	—	3	2	no	—
342	#103	—	3	2	no	—
343	#103	—	3	2	no	—
344	#102	—	3	1	no	—
345	C₂²⋊C₈	—	3	2	no	—
346	C₄×C₈	—	3	1	no	—
347	C₂×C₃×C₅	—	2	4	no	—
348	C₂×C₃×C₅	—	8	1	no	—
349	C₂×C₃×C₅	#546	1	1	yes	characteristic
350	C₂×C₃×C₅	C₂²×SD₁₆	1	1	yes	characteristic
351	C₂×C₃×C₅	#578	1	1	yes	characteristic
352	C₂³×C₃	—	2	1	no	—
353	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
354	C₃×Q₈	—	4	2	no	—
355	C₃×D₄	—	2	1	no	—
356	C₃×D₄	—	2	1	no	—
357	C₃×D₄	—	4	4	no	—
358	C₃×D₄	C₅×SD₁₆	1	4	yes	complemented
359	C₂×C₃×C₄	—	2	2	no	—
360	C₂×C₃×C₄	—	2	2	no	—
361	C₂×C₃×C₄	—	2	1	no	—
362	C₂×C₃×C₄	—	2	2	no	—
363	C₂×C₃×C₄	—	4	4	no	—
364	C₂×C₃×C₄	—	4	2	no	—
365	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
366	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
367	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
368	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
369	C₂×Dic₃	—	2	1	no	—
370	C₂×Dic₃	—	2	1	no	—
371	C₂×Dic₃	—	2	2	no	—
372	C₂×Dic₃	—	4	4	no	—
373	Dic₆	—	2	1	no	—
374	Dic₆	—	2	2	no	—
375	Dic₆	—	2	2	no	—
376	Dic₆	—	4	4	no	—
377	Dic₆	C₅×#41	1	2	yes	complemented
378	C₃⋊C₈	—	2	1	no	—
379	C₃⋊C₈	—	2	1	no	—
380	C₃⋊C₈	—	2	2	no	—
381	C₂²×C₅	—	2	2	no	—
382	C₂²×C₅	—	2	2	no	—
383	C₂²×C₅	—	4	1	no	—
384	C₂²×C₅	—	4	1	no	—
385	C₂²×C₅	—	4	1	no	—
386	C₂²×C₅	C₂×#214	1	4	yes	—
387	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
388	C₄×C₅	—	2	2	no	—
389	C₄×C₅	—	2	1	no	—
390	C₄×C₅	—	4	1	no	—
391	C₄×C₅	—	4	2	no	—
392	C₄×C₅	—	6	2	no	—
393	C₄×C₅	—	12	2	no	—
394	C₄×C₅	—	12	1	no	—
395	C₄×C₅	C₂×#214	1	2	yes	—
396	C₄×C₅	D₃×#41	1	1	yes	characteristic
397	C₄×C₅	#1009	1	1	yes	characteristic
398	#41	—	4	4	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	3	1	no	—
401	C₂×Q₈	—	3	2	no	—
402	C₂×Q₈	—	6	4	no	—
403	C₂×Q₈	—	6	2	no	—
404	C₂×Q₈	—	12	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	C₅×#67	1	1	yes	characteristic
407	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	6	2	no	—
411	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
412	SD₁₆	—	6	2	no	—
413	SD₁₆	—	12	2	no	—
414	C₂×C₈	—	3	1	no	—
415	C₂×C₈	—	3	1	no	—
416	C₂×C₈	—	3	2	no	—
417	C₂×C₈	—	12	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	3	1	no	—
421	C₄⋊C₄	—	6	1	no	—
422	C₄⋊C₄	—	6	1	no	—
423	C₄⋊C₄	—	6	2	no	—
424	C₄⋊C₄	—	6	1	no	—
425	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
426	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
427	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	6	2	no	—
431	C₂²⋊C₄	C₅×#67	1	2	yes	—
432	C₄²	—	6	1	no	—
433	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
434	C₃×C₅	#3298	1	1	yes	characteristic, complemented, Hall
435	C₂²×C₃	—	2	2	no	—
436	C₂²×C₃	—	2	2	no	—
437	C₂²×C₃	—	4	1	no	—
438	C₂²×C₃	—	4	1	no	—
439	C₂²×C₃	—	4	1	no	—
440	C₂²×C₃	C₂×C₅×SD₁₆	1	4	yes	—
441	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
442	C₃×C₄	—	2	2	no	—
443	C₃×C₄	—	2	1	no	—
444	C₃×C₄	—	4	1	no	—
445	C₃×C₄	—	4	2	no	—
446	C₃×C₄	C₂×C₅×SD₁₆	1	2	yes	—
447	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
448	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
449	Dic₃	—	2	2	no	—
450	Dic₃	—	4	2	no	—
451	Dic₃	—	4	1	no	—
452	C₂×C₅	—	2	4	no	—
453	C₂×C₅	—	8	1	no	—
454	C₂×C₅	C₂²×#214	1	1	yes	characteristic
455	C₂×C₅	#5829	1	1	yes	characteristic
456	C₂×C₅	#5838	1	1	yes	characteristic
457	C₂³	—	2	1	no	—
458	C₂³	C₂×C₅×#67	1	2	yes	—
459	Q₈	—	3	2	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	6	1	no	—
462	Q₈	—	6	2	no	—
463	Q₈	—	6	2	no	—
464	Q₈	—	12	4	no	—
465	D₄	—	2	1	no	—
466	D₄	—	2	1	no	—
467	D₄	—	4	4	no	—
468	D₄	C₅×#214	1	4	yes	complemented
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	6	1	no	—
476	C₂×C₄	—	6	1	no	—
477	C₂×C₄	—	6	2	no	—
478	C₂×C₄	—	12	4	no	—
479	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
480	C₂×C₄	C₂×C₅×#67	1	2	yes	—
481	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
482	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
483	C₈	—	6	1	no	—
484	C₈	—	6	1	no	—
485	C₈	—	6	2	no	—
486	C₂×C₃	—	2	4	no	—
487	C₂×C₃	—	8	1	no	—
488	C₂×C₃	C₅×#546	1	1	yes	characteristic
489	C₂×C₃	C₂²×C₅×SD₁₆	1	1	yes	characteristic
490	C₂×C₃	C₅×#578	1	1	yes	characteristic
491	C₅	#83531	1	1	yes	characteristic, direct product factor, Sylow
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	1	no	—
497	C₂²	C₂×C₅×#214	1	4	yes	—
498	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
499	C₄	—	2	2	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	2	no	—
503	C₄	—	6	2	no	—
504	C₄	—	12	2	no	—
505	C₄	—	12	1	no	—
506	C₄	C₂×C₅×#214	1	2	yes	—
507	C₄	C₅×D₃×#41	1	1	yes	characteristic
508	C₄	C₅×#1009	1	1	yes	characteristic
509	C₃	C₅×#3298	1	1	yes	characteristic, complemented, Sylow
510	C₂	—	2	4	no	—
511	C₂	—	8	1	no	—
512	C₂	C₂²×C₅×#214	1	1	yes	characteristic
513	C₂	C₅×#5829	1	1	yes	characteristic
514	C₂	C₅×#5838	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting, prime radical	C₃×C₅×#547	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#547	D₃
hypocenter	C₃	C₅×#3298
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×#547

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₅×D₄×Dic₆	C₂	semidirect
3. C₅×#5094	C₂	semidirect
4. C₅×#5093	C₂	semidirect
5. C₅×#5041	C₂	semidirect
6. C₅×#5040	C₂	semidirect
7. C₅×#5019	C₂	semidirect
8. C₄×C₅×#214	C₂	semidirect
9. C₅×D₄×C₃⋊C₈	C₂	semidirect
10. C₅×#4971	C₂	semidirect
11. C₂×C₅×#810	C₂	semidirect
42. #83531	C₅	direct (N × H)
91. D₄×Dic₆	C₂×C₅	semidirect
92. #5094	C₂×C₅	semidirect
93. #5093	C₂×C₅	semidirect
94. #5041	C₂×C₅	semidirect
95. #5040	C₂×C₅	semidirect
96. #5019	C₂×C₅	semidirect
97. C₄×#214	C₂×C₅	semidirect
98. D₄×C₃⋊C₈	C₂×C₅	semidirect
99. #4971	C₂×C₅	semidirect
100. C₂×#810	C₂×C₅	semidirect
137. C₃×C₅×D₄	SD₁₆	semidirect
156. C₅×Dic₆	#41	semidirect
300. C₅×D₄	#214	semidirect
358. C₃×D₄	C₅×SD₁₆	semidirect
377. Dic₆	C₅×#41	semidirect
434. C₃×C₅	#3298	semidirect
468. D₄	C₅×#214	semidirect
491. C₅	#83531	direct (N × H)
509. C₃	C₅×#3298	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#547	2
C₅×D₄×Dic₆	2
C₅×#5094	2
C₅×#5093	2
C₅×#5041	2
C₅×#5040	2
C₅×#5019	2
C₄×C₅×#214	2
C₅×D₄×C₃⋊C₈	2
C₅×#4971	2
C₂×C₅×#810	2
C₅×#3298	3
#83531	5

Subgroups — C5×#83531 #14459103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83531 #14459103